DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 479–485

CONTACT EQUIVALENCE OF MONGE-AMPÈRE EQUATIONS WITH TRANSITIVE SYMMETRIES

DIMITRY V. TUNITSKY

Abstract. This paper deals with Monge-Ampère equations on 5-dimensional contact manifolds, i.e., M-A equations with two independent variables. If a M-A equation is in general position, then the unique affine connection can be put into correspondence with this equation in natural manner. This correspondence allows to formulate and prove a number of results on contact equivalence of M-A equations with transitive symmetry group using suitable properties of affine connections.

All objects are supposed to be smooth. All manifolds are real Hausdorff paracompact 5-dimensional connected and simply connected. For any fiber bundle G denote by Γ(G) the set of all sections of this bundle.

1. Monge-Ampère equations

Consider a contact manifold M with contact structure generated by a linear differential contact form ω_{0},

dω_{0} ⋀ dω_{0} ⋀ ω_{0} ≠ 0.

The contact ideal

I_{0} = (ω_{0}, dω_{0}) (1)

is generated by the contact form ω_{0} and it’s differential dω_{0}. A differential 2-form ω is called effective if

ω ⋀ dω_{0} ⋀ ω_{0} = 0

1991 Mathematics Subject Classification. 58G03, 58G16, 53C07, 53C30. Key words and phrases. Monge-Ampère equation, characteristic bundle, contact equivalence,

characteristic affine connection, invariance with respect to parallelism, contact symmetry, homogeneous M-A equation.

Research was partially supported by International Science Foundation and Russian Government grant N RO4300 and by Russian Fundamental Researches Fund grant N 93-01-00454.

This paper is in final form and no version of it will be submitted for publication elsewhere.

(cf. [1]). Let

I = (ω_{0}, dω_{0}, ω) (2)

be the ideal of the ring Γ(Λ^{∗}(M)) that is generated by the differential forms ω_{0}, dω_{0} and ω. An ideal of the type (2) is called a Monge-Ampère ideal. A system of interior differential equations

I = 0 (3)

is called a Monge-Ampère equation.

2. Characteristic bundles

Consider the complex tangent bundle

T^{C}M = C ⊗_{R} T M

of the manifold M. A complex vector ξ ∈ T_{m}^{C}M is called a characteristic vector of the equation (3) at a point m ∈ M if ξ⌟ω_{0} = 0 and ξ⌟I(m) = ξ⌟I_{0}(m) for the ideals (1) and (2). Put

H = ⋃_{m∈M}H(m),

where

H(m) = {ξ ∈ T_{m}^{C}M : ξ⌟ω_{0} = 0, ξ⌟I(m) = ξ⌟I_{0}(m)} . (4)

Lemma 1. For any equation ( 3) there exist the two-dimensional vector bundles H_{1} and H_{2} of the complex tangent bundle T^{C}M such that the set ( 4) can be represented in the form

H(m) = H_{1}(m) ⋃ H_{2}(m)

for all points m ∈ M.

Proof. See [2].

The fiber bundles H_{1} and H_{2} are called characteristic bundles of the equation (3). This equation is called elliptic at a point m ∈ M if

H_{1}(m) ≠ H_{2}(m), \overline{H}_{1}(m) = H_{2}(m),

parabolic if

H_{1}(m) = H_{2}(m),

and hyperbolic if

H_{1}(m) ≠ H_{2}(m), \overline{H}_{j}(m) = H_{j}(m)

for j = 1, 2.

Remark 1. It can be shown that this definition of equation’s (3) type coincides with traditional one (cf. [3, ch. 5, app. 1, par. 2]).

Fix a point m ∈ M and consider two sections ξ_{j1} and ξ_{j2} of the equation’s (3) bundle H_{j} that are linearly independent at m.

Definition 1. Suppose the equation (3) is elliptic or hyperbolic at the point m. This equation is in general position at the point m if the vectors

ξ_{j1}, ξ_{j2}, [ξ_{j1}, ξ_{j2}] , [ξ_{j1}, [ξ_{j1}, ξ_{j2}]] , [ξ_{j2}, [ξ_{j1}, ξ_{j2}]]

are linearly independent at m for j = 1, 2.

3. Contact equivalence

Consider another contact manifold M̃ with contact form ω̃_{0}, Monge-Ampère ideal

Ĩ = (ω̃_{0}, dω̃_{0}, ω̃) (5)

and Monge-Ampère equation

Ĩ = 0, (6)

where ω̃ is an effective differential 2-form.

Monge-Ampère equations (3) and (6) are called contactly equivalent if there exists a diffeomorphism

f : M̃ → M (7)

that takes the ideal (2) to the ideal (5):

f^{∗}I = Ĩ. (8)

Remark 2. It follows from (8) that (7) is a contact diffeomorphism, i.e.,

f^{∗}I_{0} = Ĩ_{0}.

4. Characteristic connections

Below all Monge-Ampère equations are in general position.

Consider a Monge-Ampère equation (3) and it’s characteristic bundles H_{1} and H_{2}, which are described by lemma 1. The first derivatives H_{1}^{(1)} and H_{2}^{(1)} of these bundles are 3-dimensional complex vector subbundles of the tangent bundle T^{C}M. That is why we have the representation

T^{C}M = H_{0} ⊕ H_{1} ⊕ H_{2}, (9)

where

H_{0} = H_{1}^{(1)} ⋂ H_{2}^{(1)}

is a 1-dimensional bundle. This representation determines projections

χ^{l} : T^{C}M → H_{l}, (10)

l = 0, 1, 2, onto the corresponding summands at the right side of the representation (9).

Theorem 1. Any Monge-Ampère equation ( 3) on the contact manifold M uniquely defines the affine connection Σ on M such that the following propositions hold for this connection.

(a) For two arbitrary points m, m̃ ∈ M and for any continuous curve that connects these points the parallel displacement

F : T_{m}M → T_{m}M (11)

along this curve takes the equation’s ( 3) set H_{1} ⋃ H_{2} of all characteristic vectors at m̃ to the set of all characteristic vectors at m:

F (H_{1}(m̃) ⋃ H_{2}(m̃)) = H_{1}(m) ⋃ H_{2}(m). (12)

(b) Let ( 6) be a Monge-Ampère equation on the contact manifold M̃, Σ̃ its affine connection, and ( 7) a diffeomorphism. This diffeomorphism satisfies equality ( 8) if and only if it is affine, i.e.,

f_{∗}(Σ̃) = Σ (13)

and for some points m ∈ M and m̃ ∈ M̃ such that

m = f(m̃) (14)

it takes the equation’s ( 6) set H̃_{1} ⋃ H̃_{2} of all characteristic vectors at m̃ to the equation’s ( 3) set H_{1} ⋃ H_{2} of all characteristic vectors at m:

f_{∗}(H̃_{1}(m̃) ⋃ H̃_{2}(m̃)) = H_{1}(m) ⋃ H_{2}(m). (15)

Proof. A construction of the stated affine connection Σ is based on the representation (9). This connection is the direct sum

Σ = Σ_{0} ⊕ Σ_{1} ⊕ Σ_{2} (16)

of linear connections Σ_{0}, Σ_{1} and Σ_{2} on the bundles H_{0}, H_{1} and H_{2} respectively. The definition of these connections is the following.

The covariant differentiation operator ∇^{j} of the linear connection Σ_{j}, j = 1, 2, is uniquely defined by the properties:

∇_{X}^{j}Y = χ^{j} ([X, Y])

for X ∈ Γ(H_{0} ⊕ H_{3−j}), Y ∈ Γ(H_{j}) and

R^{j}(X, Y) = 0

for X ∈ Γ(H_{0}), Y ∈ Γ(H_{3−j}), where R^{j} is the curvature of Σ_{j} and χ^{j} is the projection operator (10).

The linear connection Σ_{0} on the bundle H_{0} is a mean of linear connections Σ_{0,1} and Σ_{0,2}:

Σ_{0} = \frac{1}{2}(Σ_{0,1} + Σ_{0,2}). (17)

The covariant differentiation operators ∇^{0,1} and ∇^{0,2} of the linear connections Σ_{0,1} and Σ_{0,2} are uniquely defined by the properties:

∇_{X}^{0,j}Y = χ^{0} ([X, Y]) (18)

for X ∈ Γ(H_{1} ⊕ H_{2}), Y ∈ Γ(H_{0}) and

R^{0,j}(X, Y) = 0 (19)

for X, Y ∈ Γ(H_{j}), where R^{0,j} is the curvature of Σ_{0,j}, j = 1, 2.

The proof of propositions (a) and (b) follows immediately from the described construction of the affine connection Σ. See [4] for technique details.

Remark 3. The construction of connections (17) and (16) implies reality of these connections.

Definition 2. The affine connection that is uniquely put into correspondence with any Monge-Ampère equation in general position according to theorem 1 is called a characteristic affine connection of this equation.

Remark 4. It can be seen from the theorem’s 2 proof that the construction of characteristic connections is based on the property of the equations under consideration to be in general position. Nevertheless the definition of the linear connection Σ_{0} on H_{0} by formulas (17), (18), and (19) is well-posed for all hyperbolic and elliptic Monge-Ampère equations. In particular, the curvature of this connection is a differential 2-form on M.

5. Equivalence theorems

Definition 3. A Monge-Ampère equation (3) in general position is invariant with respect to parallelism if the following property holds for the characteristic affine connection Σ of this equation and two arbitrary points m, m̃ ∈ M. For any continuous curve that connects these points and the parallel displacement (11) along this curve there exist neighborhoods U of m, Ũ of m̃ and a diffeomorphism

f : Ũ → U (20)

that satisfies equalities (14),

f_{∗}(m) = F,

and

f^{∗}(I | U) = I | Ũ. (21)

Theorem 2. Consider a Monge-Ampère equation ( 3) in general position and the characteristic affine connection Σ of this equation. By ∇, R and T denote the covariant differentiation, tensors of curvature and torsion of the connection Σ. The equation ( 3) is invariant with respect to parallelism if and only if the equalities

∇R = 0, ∇T = 0 (22)

hold at all points of manifold M.

Proof. Follows from theorem 1 and the well known properties of affine connections [5, ch. 6, par. 7].

Theorem 3. Consider Monge-Ampère equations ( 3) and ( 6) in general position. Denote by Σ and Σ̃ the characteristic affine connections of these equations. If the considered equations are invariant with respect to parallelism, then the following propositions are equivalent.

(a) For any points m ∈ M and m̃ ∈ M̃ there exist neighborhoods U of m, Ũ of m̃ and a diffeomorphism ( 20) such that equalities ( 14) and ( 21) hold.

(b) There exists at least one point m ∈ M and one point m̃ ∈ M̃ such that there is a linear isomorphism ( 11) that satisfies condition ( 12) and takes the connection’s Σ̃ tensors of curvature R̃(m̃) and torsion T̃(m̃) into the connection’s Σ tensors of curvature R(m) and torsion T (m) respectively.

Proof. Follows from theorem 2 and the well known properties of affine connections [5, ch. 6, par. 7].

Remind that a diffeomorphism

f : M → M

is a contact symmetry of a Monge-Ampère equation (3) on the contact manifold M if

f^{∗}(I) = I.

An equation (3) is called homogeneous if this equation’s group of contact symmetries is a Lie group that acts transitively on M. Thus M is homogeneous. If the homogeneous manifold M is reductive, then an equation (3) is called a homogeneous reductive Monge-Ampère equation.

Theorem 4. Consider a Monge-Ampère equation ( 3) in general position on the contact manifold M and the characteristic affine connection Σ of this equation. Denote by ∇, R and T the covariant differentiation, tensors of curvature and torsion of the connection Σ. The following propositions are equivalent.

(a) The equation ( 3) is a homogeneous reductive Monge-Ampère equation and Σ is a canonical connection of the manifold M.

(b) The connection Σ is complete and equalities ( 22) hold at all points of M.

Proof. Follows from theorem 1 and the well known properties of invariant connections on homogeneous reductive manifolds [5, ch. 10, par. 2].

Theorem 5. Consider Monge-Ampère equations ( 3) and ( 6) in general position and the characteristic affine connections Σ and Σ̃ of these equations. Suppose these equations satisfy proposition (a) of theorem 4. Then the following propositions are equivalent.

(a) For any points m ∈ M and m̃ ∈ M̃ there exists a diffeomorphism ( 7) such that equalities ( 14) and ( 8) hold.

(b) Proposition (b) of theorem 3.

Proof. Follows from theorem 4 and the well known properties of affine connections [5, ch. 6, par. 7].

References

1. V.V. Lychagin. Contact geometry and nonlinear second-order differential equations. Russian Math. Surv. V. 34(1) (1979), pp. 149-180.

2. T. Morimoto. La géométrie des équations de Monge-Ampère. C. R. Acad. Sc. Paris. T. 289 (1979), no 1, pp. 25-28.

3. R. Courant. Partial differential equations, Interscience, New York - London, 1962.

4. D.V. Tunitsky. Equivalence and characteristic connections of Monge-Ampère equations. To be published in Matematicheskii Sbornik (Russian).

5. S. Kobayashi, K. Nomizu. Foundations of differential geometry, V. 1, 2, Interscience, New York - London, 1963, 1969.

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